Question

Let $A=\{1,2\},B=\{3,4\}$ and $C=\{4,5\}$. Verify that $(A\times B)\times C=A\times (B\times C)$ and hence find $A\times B\times C$

Answer 1

We have

$A\times B=\{1,2\}\times \{3,4\}=\left\{\right(1,3),(1,4),(2,3),(2,4\left)\right\}$

$\Rightarrow (A\times B)\times C=\left\{\right(1,3),(1,4),(2,3),(2,4\left)\right\}\times \{4,5\}$

$=\left\{\right(1,3,4),(1,3,5),(1,4,4),(1,4,5),(2,3,4),(2,3,5),(2,4,4),(2,4,5\left)\right\}$

Again, $B\times C=\{3,4\}\times \{4,5\}=\left\{\right(3,4),(3,5),(4,4),(4,5\left)\right\}$

$\Rightarrow A\times (B\times C)=\{1,2\}\times \left\{\right(3,4),(3,5),(4,4),(4,5\left)\right\}$

$=\left\{\right(1,3,4),(1,3,5),(1,4,4),(1,4,5),(2,3,4),(2,3,5),(2,4,4),(2,4,5\left)\right\}$

$\therefore (A\times B)\times C=A\times (B\times C)=A\times B\times C$

Hence, $(A\times B\times C)=\left\{\right(1,3,4),(1,3,5),(1,4,4),(1,4,5),(2,3,4),(2,3,5),(2,4,4),(2,4,5\left)\right\}$